Problem 71 - Precalculus with Limits

Question

PaTH OF A BALL The path of a ball thrown by a child is modeled by $$ y=-x^{2}+5 x+2 $$ where \(y\) is the height of the ball (in feet) and \(x\) is the horizontal distance (in feet) from the point from which the ball was thrown. Using your knowledge of the slopes of tangent lines, show that the height of the ball is increasing on the interval \([0,2]\) and decreasing on the interval \([3,5]\) . Explain your reasoning.

Step-by-Step Solution

Answer

Therefore, it can be demonstrated that the height of the ball is increasing on the interval [0, 2] and decreasing on the interval [3, 5].

Step 1: Find the derivative

Firstly, find the derivative of the function \(y = -x^2 + 5x + 2\) to find the gradient or slope and therefore deduce whether it's increasing or decreasing. The derivative is given by \[y' = f'(x) = -2x+5\].

Step 2: Check for the interval [0, 2]

Within the range [0, 2], plug in \(x = 1\) (which lies in the middle of the interval [0, 2]) into the derivative function. The result is \((y'(1) = -2(1) + 5 = -2 + 5 = 3)\). Since we get a positive result when the value of \(x\) in this given range is substituted into the derivative, this means that the function \(y\) is increasing in the interval [0, 2].

Step 3: Check for the interval [3, 5]

Next, within the range [3, 5], plug in \(x = 4\) (which lies in the middle of the interval [3, 5]) into the derivative function. The result is \(y'(4) = -2(4)+5 = -8 + 5 = -3\). Since we get a negative result when the value of \(x\) in this given range is substituted into the derivative, this means that the function \(y\) is decreasing in the interval [3, 5].